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Group action analysis developed and applied mainly by Louis Michel to the study of N-dimensional
periodic lattices is the central subject of the book. Different basic mathematical tools
currently used for the description of lattice geometry are introduced and illustrated through
applications to crystal structures in two- and three-dimensional space, to abstract multi-dimensional
lattices and to lattices associated with integrable dynamical systems. Starting from general Delone
sets the authors turn to different symmetry and topological classifications including explicit construction
of orbifolds for two- and three-dimensional point and space groups.
Voronoï and Delone celles together with positive quadratic forms and lattice description by root
systems are introduced to demonstrate alternative approaches to lattice geometry study. Zonotopes
and zonohedral families of 2-, 3-, 4-, 5-dimensional lattices are explicitly visualized using
graph theory approach. Along with crystallographic applications, qualitative features of lattices of
quantum states appearing for quantum problems associated with classical Hamiltonian integrable
dynamical systems are shortly discussed.
The presentation of the material is presented through a number of concrete examples with an extensive
use of graphical visual zation. The book is aimed at graduated and post-graduate students and
young researchers in theoretical physics, dynamical systems, applied mathematics, solid state physics,
crystallography, molecular physics, theoretical chemistry, ...
Book series edited by Michèle Leduc and Michel Le Bellac.
